2,598 research outputs found
Stable, Robust and Super Fast Reconstruction of Tensors Using Multi-Way Projections
In the framework of multidimensional Compressed Sensing (CS), we introduce an
analytical reconstruction formula that allows one to recover an th-order
data tensor
from a reduced set of multi-way compressive measurements by exploiting its low
multilinear-rank structure. Moreover, we show that, an interesting property of
multi-way measurements allows us to build the reconstruction based on
compressive linear measurements taken only in two selected modes, independently
of the tensor order . In addition, it is proved that, in the matrix case and
in a particular case with rd-order tensors where the same 2D sensor operator
is applied to all mode-3 slices, the proposed reconstruction
is stable in the sense that the approximation
error is comparable to the one provided by the best low-multilinear-rank
approximation, where is a threshold parameter that controls the
approximation error. Through the analysis of the upper bound of the
approximation error we show that, in the 2D case, an optimal value for the
threshold parameter exists, which is confirmed by our
simulation results. On the other hand, our experiments on 3D datasets show that
very good reconstructions are obtained using , which means that this
parameter does not need to be tuned. Our extensive simulation results
demonstrate the stability and robustness of the method when it is applied to
real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity
based CS methods specialized for multidimensional signals is also included. A
very attractive characteristic of the proposed method is that it provides a
direct computation, i.e. it is non-iterative in contrast to all existing
sparsity based CS algorithms, thus providing super fast computations, even for
large datasets.Comment: Submitted to IEEE Transactions on Signal Processin
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
Sampling in the Analysis Transform Domain
Many signal and image processing applications have benefited remarkably from
the fact that the underlying signals reside in a low dimensional subspace. One
of the main models for such a low dimensionality is the sparsity one. Within
this framework there are two main options for the sparse modeling: the
synthesis and the analysis ones, where the first is considered the standard
paradigm for which much more research has been dedicated. In it the signals are
assumed to have a sparse representation under a given dictionary. On the other
hand, in the analysis approach the sparsity is measured in the coefficients of
the signal after applying a certain transformation, the analysis dictionary, on
it. Though several algorithms with some theory have been developed for this
framework, they are outnumbered by the ones proposed for the synthesis
methodology.
Given that the analysis dictionary is either a frame or the two dimensional
finite difference operator, we propose a new sampling scheme for signals from
the analysis model that allows to recover them from their samples using any
existing algorithm from the synthesis model. The advantage of this new sampling
strategy is that it makes the existing synthesis methods with their theory also
available for signals from the analysis framework.Comment: 13 Pages, 2 figure
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